in the spatial coordinates
at time , where
and
are
non-singular, non-negative matrices which play the respective roles of
multidimensional mass and stiffness. The second spatial derivative is
defined here as

where is interpreted as a Laplace-transform variable
,
is the identity matrix,
,
is a diagonal
matrix of spatial
Laplace-transform variables (the imaginary part of being
spatial frequency along the th spatial coordinate), and
. Substituting the eigenfunction (7) into
(5) gives the algebraic equation

(8)

where
is the diagonal matrix of sound-speeds along the coordinate axes. Since
, we have

(9)

Substituting (9) into (7),
the eigensolutions of (5) are found to be of the form

(10)

Having established that (10) is a solution of
(5) when condition (8) holds for the matrices
and
, we can express the general traveling-wave
solution to (5) in both pressure and velocity as

(11)

where
, with being an arbitrary
superposition of right-going components of the form (10)
(i.e., taking the minus sign), and
is similarly any linear combination of left-going eigensolutions from
(10) (all having the plus sign). Similar definitions apply for
and
.
When the time and space arguments are dropped as in the right-hand side of
(11), it is understood that all the quantities are
written for time and position
.

When the mass and stiffness matrices
and
are diagonal, our analysis corresponds to considering separate
waveguides as a whole. For example, the three directions of vibration
(one longitudinal and two transverse) in
a single terminated string can be described by
(5) with . The coupling among the strings occurs
primarily at the bridge in a piano [132]. As we will see
later, the bridge acts like a junction of several multivariable
waveguides.

When the matrices
and
are
non-diagonal, the physical interpretation can be of the form

(12)

where
is the stiffness matrix, is the
mass density matrix.
is diagonal if (8) holds, and in this case, the wave
equation (5) is decoupled in the spatial dimensions.
There are physical examples where the matrices
and
are not diagonal, even though
is.
One such example, in the domain of
electrical variables, is given by conductors in a sheath or above
a ground plane, where the sheath or the ground plane acts as a
coupling element [63, pp. 67-68].

Note that the multivariable wave equation (5) considered here
does not include wave equations governing propagation in multidimensional
media (such as membranes, spaces, and solids). In higher dimensions, the
solution in the ideal linear lossless case is a superposition of waves
traveling in all directions in the -dimensional
space [60]. However, it turns out [122]
that a good simulation of wave
propagation in a multidimensional medium may be in fact be obtained by
forming a mesh of unidirectional waveguides as considered here, each
described by (5). Such a mesh of 1D
waveguides can be shown to solve numerically a discretized wave equation
for multidimensional media [125].